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Sparse PCA Beyond Covariance Thresholding
\] Prior to this work, the best polynomial time algorithm in the regime $k\approx \sqrt{d}$, called \emph{Covariance Thresholding} (proposed in [KNV15a] and analyzed in [DM14]), required $\beta \gtrsim \frac{k}{\sqrt{n}}\sqrt{\ln({2 + d/k^2})}$.
Higher degree sum-of-squares relaxations robust against oblivious outliers
Using a reduction from the planted clique problem, we provide evidence that the quasipolynomial time is likely to be necessary for sparse PCA with symmetric noise.
Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers
For sparse regression, we achieve consistency for optimal sample size $n\gtrsim (k\log d)/\alpha^2$ and optimal error rate $O(\sqrt{(k\log d)/(n\cdot \alpha^2)})$ where $n$ is the number of observations, $d$ is the number of dimensions and $k$ is the sparsity of the parameter vector, allowing the fraction of inliers to be inverse-polynomial in the number of samples.
Sparse PCA: Algorithms, Adversarial Perturbations and Certificates
Despite a long history of prior works, including explicit studies of perturbation resilience, the best known algorithmic guarantees for Sparse PCA are fragile and break down under small adversarial perturbations.
Consistent regression when oblivious outliers overwhelm
Concretely, we show that the Huber loss estimator is consistent for every sample size $n= \omega(d/\alpha^2)$ and achieves an error rate of $O(d/\alpha^2n)^{1/2}$.
